A237260 Least positive integer m < n with prime(prime(m)) + 2 and prime(n-m) + 2 both prime, or 0 if such a number m does not exist.

0, 0, 1, 1, 2, 1, 2, 1, 2, 3, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 1, 2, 3, 4, 23, 6, 1, 2, 1, 2, 3, 4, 7, 1, 2, 1, 2, 3, 4, 7, 6, 1, 2, 1, 2, 1, 2, 3, 4, 1, 2, 3, 1, 2, 3, 4, 14, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 24, 1

Offset

1,5

Comments

Conjecture: a(n) < sqrt(6*n)*log(3*n) for all n > 0.

We have verified this for n up to 5*10^5. Note that a(273) = 271 > sqrt(6*273)*log(2*273).

According to the conjecture in A218829, a(n) should be positive for all n > 2.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Example

a(5) = 2 since prime(prime(2)) + 2 = prime(3) + 2 = 7 and prime(5-2) + 2 = 7 are both prime, but prime(5-1) + 2 = 7 + 2 = 9 is composite.

Mathematica

pq[k_, m_]:=PrimeQ[Prime[k]+2]&&PrimeQ[Prime[Prime[m]]+2]
Do[Do[If[pq[n-m, m], Print[n, " ", m]; Goto[aa]], {m, 1, n-1}];
Print[n, " ", 0]; Label[aa]; Continue, {n, 1, 70}]

Crossrefs

Cf. A000040, A001359, A006512, A218829, A237259.

Sequence in context: A136314 A121997 A023128 * A249727 A023118 A122197

Adjacent sequences: A237257 A237258 A237259 * A237261 A237262 A237263

Keyword

nonn

Author

Zhi-Wei Sun, Feb 05 2014

Status

approved